Classification of injective factors. Cases \(\mathrm{II}_1\), \(\mathrm{II}_\infty\), \(\mathrm{III}_\lambda\), \(\lambda\neq 1\)
type: Item
instance of: scholarly article
title: Classification of injective factors. Cases \(\mathrm{II}_1\), \(\mathrm{II}_\infty\), \(\mathrm{III}_\lambda\), \(\lambda\neq 1\)
zbMATH Open document ID: 0343.46042
DOI: 10.2307/1971057
author: Alain Connes
published in: Annals of Mathematics. Second Series
publication date: 1976-01-01T00:00:00Z
review text: The paper contains definitive results an hyperfiniteness and injectivity of von Neumann algebras, which give the solutions of many important problems in the theory of operator algebras. Let \(N\) be a von Neumann algebra on a Hilbert space \(H\) and \(B(H)\) the algebra of all bounded linear operators in \(H\). \(N\) is said to be injective if there is a projection of norm one of \(B(H)\) to \(N\) or equivalently if, for a \(C^*\) algebra \(A\) and its \(C^*\)-subalgebra \(B\), any completely positive map of \(B\) into \(N\) has a completely positive extension to \(A\) [\textit{J. Hakeda} and the reviewer, Tĥoku math. J., II. Ser. 19, 315--323 (1967; Zbl 0175.14201); \textit{E. Effros} and \textit{C. Lanee}, Tensor products of operator algebras, to appear in Advances Math.]. The algebra \(N\) is also said to be semidiscrete if the identity map \(N\to N\) is approximated in \(\sigma\)-weak topology by a net of completely positive maps of finite rank. The author's main result asserts that for a factor \(N\) of type II\(_1\) in a separable Hilbert space the notions of injectivity and semidiscreteness are equivalent to the hyperfiniteness of \(N\), the weak closure of an ascending sequence of matrix algebras (results are stated in separated theorems). He also proved further equivalence of these properties to those of the property \(P\) by \textit{J. T. Schwartz} [Commun. Pure Appl. Math. 16, 19--26 (1963; Zbl 0131.33201)] and the property \(\Gamma\) [\textit{F. J. Murray} and \textit{J. von Neumann}, Ann. Math. (2) 44, 716--808 (1943; Zbl 0060.26903)]. Thus, as natural consequences of these results one knows that up to isomorphisms there is only one injective factor of type II\(_1\), a hyperfinite factor and the hyperfinite factor of type II\(_\infty\) is unique. It is also now clear that all subfactors of a hyperfinite factor \(R\) of type III\(_1\) are isomorphic to \(R\) or finite dimensional. The equivalences of those properties are further shown to be valid for any…
Mathematics Subject Classification ID: 46L10
Mathematics Subject Classification ID: 46M10
zbMATH DE Number: 3535135
Wikidata QID: Q56213148
reviewed by: Jun Tomiyama
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/1971057
OpenAlex ID: W2328912295
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description: scientific article; zbMATH DE number 3535135